Quadratic Primes
N. A. Carella
Abstract: The subset of quadratic primes  p = a n2 + b n + c : n ∈ ℤ  generated by some irreducible polynomial
f (x) = a x2 + b x + c ∈ ℤ[x] over the integers ℤ is widely believed to be an unbounded subset of prime numbers. This work
provides the details of a possible proof for some quadratic polynomials f (x) = x2 + d, 1 ≤ d ≤ 100. In particular, it is
shown that the cardinality of the simplest subset of quadratic primes  p = n2 + 1 : n ∈ ℤ  is infinite.
Mathematics Subject Classifications: 11A41, 11N32, 11N13.
Keywords: Distribution of Primes; Quadratic Primes Conjecture; Landau Prime Problem; Bouniakowsky Conjecture;
Elliptic Twin Primes Conjecture; Prime Diophantine Equations.
.
1. Introduction
Let f (x) = a x2 + b x + c ∈ ℤ[x] be an irreducible polynomial over the integers ℤ = { ..., -3, -2, -1, 0, 1, 2, 3, ... }.
The quadratic primes conjecture claims that certain Diophantine equation y = a x2 + b x + c has infinitely many prime
solutions y = p as the integers n ∈ ℤ varies over the set of integers. More generally, the Bouniakowsky conjecture claims
that for an irreducible f (x) ∈ ℤ[x] over the integers of fixed divisor div( f ) = 1, and degree deg( f ) ≥ 2, the Diophantine
equation y = f (x) has infinitely many prime solutions y = p as the integer n ∈ ℤ varies over the set of integers. The fixed
divisor div( f ) = gcd( f (ℤ)) of an irreducible polynomial f (x) ∈ ℤ[x] over the integers is the greatest common divisors of
its image f (ℤ) = { f (n) : n ∈ ℤ } over the integers. The fixed divisor div( f ) = 1 if the congruence f (n) ≡ 0 mod p has
w( p) < p solutions for all prime numbers p ≤ deg( f ), see [FI10, p. 395]. Detailed discussions of the quadratic primes
conjecture appear in [RN96, p. 387], [LG10, p. 17], [FI10, p. 395], [NW00, p. 405], [PJ09, p. 33], [IH78], [DI82], [LR12],
[HW08], and related topics in [BZ07], [CC00], [GM00], [MK09], et alii.
This note provides the details of a possible proof for some quadratic polynomials f (x) = x2 + d, 1 ≤ d ≤ 100. In particular,
the cardinality of the simplest subset of quadratic primes  p = n2 + 1 : n ∈ ℤ  is infinite. The techniques employed here
are much simpler than the standard sieve methods employed in [IH78], [DI82], and [LR12], and a few other authors. This
analysis is based on a simple weighted sieve. Essentially, it is a synthesis of those techniques used in [HB10, p. 1- 4], and
by other authors.
Theorem 1.
Let f (x) = x2 + d ∈ ℤ[x] be an irreducible polynomial over the integers of fixed divisor div( f ) = 1, and
1 ≤ d ≤ 100. Then, the Diophantine equation p = n2 + d has infinitely many primes solutions y = p as n ∈ ℤ varies over
the integers.
2 | Quadratic Primes
Proof: Without loss in generality, let f (x) = x2 + 1. Since the congruence n2 + 1 ≡ 0 mod p has less than p solutions for
any prime p ≤ deg( f ) = 2, the fixed divisor of the polynomial is div( f ) = 1. Now, select an appropriate weighted finite
sum over the integers as observed in (16):
Λn2 + 1
1
=- 

n2 +1 ≤ x
log n
n
n2 +1 ≤ x
 μ(d) log d
log n
n
d  n2 +1
= -  μ(d) log d
.

n
n2 +1 ≤ x,
n2 +1≡0 mod d
d≤x
(1)
1
log n
where Λ(n) = - ∑d n μ(d) log d, and gcd(d, n) = 1. This follows from Lemma 2, and inverting the order of summation.
Other examples of inverting the summation are given in [MV07, p. 35], [RH94, p. 27], [RM08, p. 216], [SN83, p. 83],
[TM95, p. 36], and other.
Since the small moduli d ≥ 1 contribute the bulk of the main term of the finite sum (1), and the large moduli have insignificant contribution, the last finite sum is broken up into two finite sums according to d ≤ x1/2-ϵ or x1/2-ϵ < d ≤ x, where
ϵ > 0 is an arbitrarily small number. Specifically, the dyadic decomposition has the form
Λn2 + 1

n2 +1 ≤ x
n
1
= -  μ(d) log d
log n
d≤x1/2-ϵ

log n
n
n2 +1 ≤ x,
n2 +1≡0 mod d
-
1
μ(d) log d

x1/2-ϵ <d≤x

n2 +1 ≤ x,
n2 +1≡0 mod d
.
log n
n
(2)
Small Moduli d ≤ x1/2-ϵ , ϵ > 0: Applying Lemmas 4 and 7 yield
1
-  μ(d) log d
⩾ -  μ(d) log d c0

d≤x1/2-ϵ
n2 +1 ≤ x,
n
log n
d≤x1/2-ϵ
ρ(d)
d
log x 1 + O
1
xϵ log x
n2 +1≡0 mod q
≥ -c0
log x
μ(d) ρ(d) log d

d
d≤x1/2-ϵ
≥ c0
log x c1 + Oe
-c
log x

log x
xϵ
 + O
ρ(d) log d
1
+O
log2 x
xϵ/2
d≤x1/2-ϵ
,
where ρ(q) =   n ≤ x1/2 : n2 + 1 ≡ 0 mod q  = Oxϵ/2 , see (11), and c0 =
constants.
(3)
d
2 -
2 (1 - ϵ) > 0, c1 > 0, c > 0 are
Quadratic Primes | 3
Large Moduli d > x1/2-ϵ , ϵ > 0: Applying Lemmas 5 and 6 yield
-

μ(d) log d
x1/2-ϵ <d≤x
1
-c
= Oe

n2 +1 ≤ x,
n
log x
4
log x + O
log n
log4 x
x(loglog x) (logloglog x)/log x
.
(4)
n2 +1≡0 mod q
Combining these expressions into (2) yield
Λn2 + 1
≫

n2 +1 ≤ x
n
log x 1 + O
log n
log7/2 x
x(loglog x) (logloglog x)/log x
.
(5)
Since the contribution by the subset of prime powers n2 + 1 = pv ≤ x, v ≥ 2, is zero, that is,
Λn2 + 1

n2 +1= pv ≤ x, v≥2
n
log n
= 0,
(6)
see Lemmas 12 and 13 in Section 7, it follows that the cardinality of the subset of primes  n 2 + 1 = p ≤ x, n ∈ ℤ  is
unbounded as x⟶∞.
■
Some irreducible polynomials f (x) = a x2 + b x + c ∈ ℤ[x] of fixed divisor div( f ) = 1 can be transformed into an equivalent case of the form g(x) = x2 + d by means of algebraic manipulations. The equivalent problem is then handled as the
case f (x) = x2 + 1 presented above. Moreover, a lower bound of the asymptotic formula is provided in Section 8.
The topics of primes in quadratic, cubic, quartic arithmetic progressions, the Bouniakowsky conjecture, and in general the
Hypothesis H, and the Bateman-Horn conjecture, [RN96], are rich areas of research involving class fields theory and
analytic number theory. The linear case, Dirichlet Theorem for primes in arithmetic progressions, is proved in [CS09,
Chapter 2] from the point of view of class fields theory. The quadratic case, Theorem 1 here, seems to have another proof
in term of Hecke L-functions over the Gaussian quadratic field  -1  quite similar to Dirichlet Theorem's proof in term
of L-functions over the rational field .
As an application, it can be shown that there are irreducible quadratic polynomials f (x) = a x 2 + b x + c ∈ ℤ[x] of fixed
divisor div( f ) ≠ 1 such that f (n) = P2 (n) is the product of two primes for infinitely many values. For example, take the
irreducible polynomial f (x) = x(x + 1) + 2 of fixed divisor div( f ) = 2. By Theorem 1, f (n) / div( f ) = p is prime for
infinitely many values. Ergo, f (n) = 22 p for infinitely many values n ∈ ℤ. The topic of almost primes is studied in [IH78],
[DI82], and [LR12], using sieve methods.
The elementary underpinning of Theorem 1 is assembled in Sections 2 to 8. In Lemma 14, a lower bound of the correspond-
4 | Quadratic Primes
ing primes counting function is computed. The proofs of all these lemmas use elementary methods.
2. Elementary Foundation
The basic definitions of several number theoretical functions, and a handful of Lemmas are recorded here.
2.1 Formulae for the Mobius and vonMangoldt Functions
Let n ∈ ℕ = {0, 1, 2, 3, ... } be an integer. The Mobius function is defined by
μ(n) =
v
v
if n = p11 · p22 ⋯ ptvt , if v1 = ⋯ = vt = 1,
(-1)t
0
if n =
v
p11
·
v
p22
⋯
ptvt ,
(7)
if some vi ≠ 1.
The subset of squarefree numbers { n ∈ ℕ : μ(n) ± 1 } = { n = p1 · p2 ⋯ pt : pi prime } is the support of the Mobius function
μ : ℕ ⟶{ -1, 0, 1 }. Further, the vonMangoldt function is defined by
Λ(n) =
log p
if n = pk , k ≥ 1,
0
if n ≠ p , k ≥ 1.
(8)
k
The subset of prime powers { n ∈ ℕ : Λ(n) ≠ 0 } =  n ∈ ℕ : n = pk , k ≥ 1  is the support of the vonMangoldt function
Λ : ℕ ⟶.
Lemma 2. Let n ≥ 1 be an integer, and let Λ be the vonMangoldt function. Then
Λ(n) = -  μ(d) log d
(9)
d n
Proof: Use Mobius inversion formula
f (n) =  g(d)
⟺
g(n) =  μ(d) f (n / d)
d n
(10)
d n
on the identity log n = ∑d n Λ(d) to confirm this claim. 
Extensive details for other identities and approximations of the vonMangoldt are discussed in [GP05], [FI10], [HN07, p.
27], et alii.
2.2 Quadratic Equations Over Arithmetic Progressions
Let 1 ≤ a < q be integers, gcd(a, q) = 1. An element a ∈ { 0, 1, 2, 3, ... q - 1 } ≅ ℤ / q ℤ is called a quadratic residue if
the congruence z2 ≡ a mod q has a solution. Otherwise, a ≥ 1 is a quadratic nonresidue. For a quadratic residue a modulo q,
with gcd(a, q) = 1, the congruence z2 ≡ a mod q has 2W ≥ 2 solutions, where W = ω (q) + r, r = 0, 1, 2, according as
Quadratic Primes | 5
4  q, or 4  q or 8  q, see (11) and [LV56, p. 65]. The function ω(q) tallies the number of prime divisors of q, cf [CC07]
Lemma 3. (Fermat) Let p ≥ 3 be a prime number. Then
(i) The integer -1 is quadratic residue modulo p if and only if p = u2 + v2 , u, v ∈ ℤ.
(ii) The integer -1 is quadratic nonresidue modulo p if and only if p ≠ u2 + v2 , u, v ∈ ℤ.
The quadratic residuacity of numbers and related concepts are explicated in almost every textbook in elementary number
theory, [HW08], [SN83], [LV56], [RH94], et alii.
The previous information quickly leads to a formula for the multiplicative function ρ(q) =   n ≤ x 1/2 : n2 + 1 ≡ 0 mod q .
For any integer q ≥ 2 this is defined by
where
 ρ( p),
ρ( p) =
pq
1
if p = 2,
2
if p = u2 + v2 ,
0
if p ≠ u2 + v2 or p = 2k , k ≥ 2.
(11)
These concepts come into play in the analysis of powers sums, and other finite sums over quadratic arithmetic progressions.
3. Finite Sums Over Small Moduli
The small moduli q = O
log x , as demonstrated below, have the largest effect and contribute the bulk of the main term
in the finite sum considered here.
Lemma 4.
Let x ∈  be a large number, and let q < x be an integer. Then,
1
(i)

2
n +1 ≤ x,
n
log n
2 ρ(q)
=

q
logx1/2 - q f (q) -
log r(q)  ,
if q < x.
n2 +1≡0 mod q
1
(ii)

n2 +1 ≤ x,
n
log n
≥ c0
ρ(q)
q
log x 1 + O
1
x log x
ϵ
,
if q ≤ x1/2-ϵ , ϵ > 0 .
n2 +1≡0 mod q
(12)
where r(q) ≥ 1 is a solution of the quadratic congruence n2 + 1 ≡ 0 mod q, the term ρ(q) =   n ≤ x1/2 : n2 + 1 ≡ 0 mod q 
is the number of solutions, the term f (q) = x1/2 - r(q)  q is the fractional part function, and c0 =
constant.
2 -
2 (1 - ϵ) is a
6 | Quadratic Primes
Proof: (i) Fix a solution r = r(q) ≥ 1 of the quadratic congruence n2 + 1 ≡ 0 mod q such that r2 + 1 ≤ x. Then, each integer
n = q m + r ≤ x1/2 is also a solution. Thus, the finite sum can be rewritten as
1
2
n +1 ≤ x,
n
1
= 

log n
m ≤V
,
log (q m + r)
(q m + r)
n2 +1≡0 mod q
(13)
where V = x1/2 - r  q - x1/2 - r  q, and r = r(q) < q < x. Evaluating the integral representation yields
1
n2 +1 ≤ x,
n
1
V
=

log n
0
(q t + r)
dt
log (q t + r)
n2 +1≡0 mod q
V
=
2
q
log (q t + r)
=
2
q

logx1/2 - q f (q) -
log r(q) ,
0
(14)
where f (q) = x1/2 - r  q is the fractional part function.
For (ii), let q ≤ x1/2-ϵ , where ϵ > 0 is an arbitrarily small number, and r(q) < q, then
logx1/2 - q f (q) -
log r(q) ≥
≥
logx1/2 - x1/2-ϵ  1/2
log x
≥  1/2 -
1+
log x1/2-ϵ
log(1 - x-ϵ )
log x1/2
1/2-ϵ 
-
log x1/2-ϵ
1
log x + O
xϵ
log x
> 0.
(15)
Substituting this back, and simplifying it, provide a lower bound of the main term. ■
Various other finite sums having unbounded partial sums appear to be suitable for this analysis. For example,
Quadratic Primes | 7
Λn2 + 1

2
n +1 ≤ x
n (log n)1-α
, α > 0.
(16)
4. Finite Sums Over Large Moduli
This Section is concerned with effective estimates of certain finite sums over large moduli. The finite sum over the subset
of large moduli, which have small number of prime divisors ω(q) ≤ loglog x, is covered in Subsection 4.1. The other finite
sum over the subset of moduli, which have large number of prime divisors ω(q) > loglog x, is covered in Subsection 4.2.
4.1 Case ω(q) ≤ loglog x
Let ω(q) be the number of prime divisors of q. Each integer in the subset of large moduli { q ≤ x : ω(q) ≤ loglog x } has a
small number of prime divisors. Accordingly, the quadratic congruence z2 ≡ a mod q has a small number of solutions
ρ(q) ≤ 2ω(q)+2 ≤ log x. The subset of integers { q ≤ x : ω(q) ≤ loglog x } has density 1 in the set of nonnegative integers
ℕ = { 0, 1, 2, 3, ... }.
Let x ∈  be a large number, and let q > x1/2-ϵ , ϵ > 0, be an integer such that ω(q) ≤ loglog x. Then,
Lemma 5.
-
1
μ(q) log q

x1/2-ϵ <q≤x,
ω(q)≤ loglog x

n
n2 +1 ≤ x,
≪ e-c
log x
log2 x.
(17)
log n
n2 +1≡0 mod q
Proof: Let r ≥ 1 be a root of n2 + 1 ≡ 0 mod q. As the moduli q are restricted to the range x1/2-ϵ < q ≤ x, and
n = q m + r ≤ x1/2 , these data imply that x1/2-ϵ m + r ≤ n = q m + r ≤ x1/2 . Hence, the index m is restricted to the range
0 ≤ m ≤ xϵ .
Apply Lemma 2, see also (1) for finer details, and rearrange the finite sum as
-

x1/2-ϵ <q≤x,
ω(q)≤ loglog x
=-

1/2-ϵ
1
μ(q) log q
x
<q≤x,
ω(q)≤ loglog x

n
n2 +1 ≤ x,
log n
2
n +1≡0 mod q
μ(q) ρ(q) log q
1

m ≤ xϵ
(q m + r)
,
log (q m + r)
(18)
where ρ(q) =   n ≤ x1/2 : n2 + 1 ≡ 0 mod q  ≤ log x. Use an integral representation to estimate the inner finite sum as
follows:
8 | Quadratic Primes
1

m ≤ xϵ
1
xϵ
=
0
log (q m + r)
(q m + r)
dt=
log(q t + r)
( q t + r)
2
q

log(q log x + r) -
log r  ,
(19)
where 1 ≤ r < q < x. Substituting these information, using summation by part, and applying Lemma 7, return
-
x1/2-ϵ <q≤x,
ω(q)≤ loglog x
=-
1
μ(q) ρ(q) log q


μ(q) ρ(q) log q
1/2-ϵ
x
<q≤x,
ω(q)≤ loglog x
2
q
μ(d) ρ(q) log q

1/2-ϵ
x
<q≤x,
ω(q)≤ loglog x
≪ e-c
log x
log n
n +1≡0 mod q

= -2
n
n2 +1 ≤ x,
2
q

log(q xϵ + r) -
log r 
log(q xϵ + r) + 2

μ(d) ρ(q) log q
1/2-ϵ
x
<q≤x,
ω(q)≤ loglog x
q
log r
log4 x,
(20)
where r = r(q), refer to Lemma 7-iii, this estimate uses ρ(q) ≪ log x. ■
The reader can confer [MV07, p. 182], and [RM08, p. 318] for similar evaluations.
4.2 Case ω(q) > loglog x
Let ω(q) be the number of prime divisors of q. Each integer in the subset of large moduli { q ≤ x : ω(q) > loglog x } has a
large number of prime divisors. Accordingly, the quadratic congruence z2 ≡ a mod q has a large number of solutions
ρ(q) ≤ 2ω(q)+2 ≤ qδ , where δ > 0 is an arbitrarily small number.
The subset { q ≤ x : ω(q) > loglog x } of such highly composite integers has zero density in the set of integers
ℕ = { 0, 1, 2, 3, ... }. In fact, this is a very small subset of integers  { q ≤ x : ω(q) > loglog x } = o(x). This topic is
discussed in Section 6, and some related information are given in [AE44, p. 449].
Lemma 6.
Let x ∈  be a large number, let q > x1/2-ϵ , ϵ > 0, be an integer, and let ω(q) > loglog x. Then,
Quadratic Primes | 9
-
≪

1/2-ϵ
log n
n
2
x
<q≤x,
ω(q)> loglog x
log4 x
1
μ(d) log d

n +1 ≤ x,
x(loglog x) (logloglog x)/log x
.
n2 +1≡0 mod q
Proof: Here the moduli q are restricted to the range x1/2-ϵ < q ≤ x, and n = q m + r ≤ x1/2 . Hence, the index m is restricted
to the range 0 ≤ m ≤ xϵ . Let r(q) ≥ 1 be a root of z2 + 1 ≡ 0 mod q. Next rearrange the finite sum as
-
x1/2-ϵ <q≤x,
ω(q)> loglog x
=-
1
μ(d) log d


μ(d) ρ(q) log d

x1/2-ϵ <q≤x,
ω(q)> loglog x
=-
μ(d) ρ(q) log d
x log x<q≤x,
ω(q)> loglog x
≪
log n
1

m ≤ xϵ

1/2
n
n2 +1 ≤ x,
n2 +1≡0 mod q
-1
2
q
ρ(q) log2 q
log xϵ

q
1/2-ϵ
x
<q≤x,
ω(q)> loglog x

(q m + r)
log(q m + r)
log(q xϵ + r) -
log r(q) 
,
(22)
refer to Lemma 4 for similar calculations. Applying Lemma 11 completes the proof. ■
5. Finite Sums of Mobius Function
Lemma 7.
Let x ∈  be a large number, and let f (n) = OlogB x, B > 0, be a function. Let q ≥ 1, a ≥ 1 be a pair of
integers, gcd(a, q) = 1. Then,
(i)

μ(d) = Ox e-c
log x
,
n≤x, n≡a mod q
μ(n) log n
(ii)

n≤x, n≡a mod q
n
= -c0 + Oe-c
μ(n) log n f (n)
(iii)

n≤x, n≡a mod q
n
= Ox e-c
log x
,
log x
(23)
f (x) log x,
10 | Quadratic Primes
where c0 > 0, c > 0 are constants.
Some of these finite sums are evaluated in [MV07, p. 182-185], [NW00, p. 351], and [RM08].
6. Highly Composite Integers
The statistics on subsets of integers with specified number of primes is of general interest in many area of mathematics.
Some information concerning these integers are recorded in this Section.
Lemma 8. (Landau) Let x ∈  be a large number, and let πk (x) =  { n ≤ x : ω(n) = k } be the counting function for the
number of integers with k-prime factors. Then,
x(loglog x)k-1
πk (x) =
(k - 1) ! (log x)
+o
x(loglog x)k-1
(k - 1) ! (log x)
.
(24)
The proof is usually done for the restricted range k ≪ loglog x, but it holds for any parameter k ≤ log x. Detailed proofs are
given in [DF12, p. 159], [ND12], and [TM95].
Lemma 9.
Let x ∈  be a large number, and let πk (x) =  { n ≤ x : ω(n) = k }, be the k-prime factors integers counting
function. Then,
πk (x) ≪ x.

loglog x≤k≤log x
(25)
Proof: By Lemma 8, this finite sum has the upper bound
x(loglog x)k-1

loglog x≤k≤log x
πk (x) ≪ 
k≤log x
(k - 1)! (log x)
≪
(loglog x)k-1
x
log x

k≥1
(k - 1) !
(26)
= x. ■
The function W 0 (t) =  { q ≤ t : loglog t ≤ ω(q) ≤ log t } accounts for the cardinality of the subset of integers with at least
k ≥ loglog t prime factors. An estimate for the closely related finite sum involving the counting measure
ρk (q) =   n ≤ x1/2 : n2 + 1 ≡ 0 mod q and ω(q) = k  is stated here.
Lemma 10.
Let x ∈  be a large number, and let πk (x) =  { n ≤ x : ω(n) = k }, be the k-prime factors integers counting
function. Then,
Quadratic Primes | 11
W (x) =
ρk (x) πk (x) ≪ x

1-
(loglog x) (logloglog x)
2 log x
.
(27)
loglog x≤k≤log x
Proof: Express the following quantities in powers of x ≥ 1:
k loglog x
(log x)k = x
log x
(k-1) logloglog x
(loglog x)k-1 = x
,
log x
,
(28)
k log k-k+O(log k)
(k - 1)! ≪ x
k+2
ρ(q) ≤ 2ω(q)+2 = 2k+2 ≤ x log x .
,
log x
Substituting the previous expressions into the finite sum upper bound of the inner finite sum, and using Lemma 8, return
ρk (x) πk (x) ≪

loglog x≤k≤log x
2

1+
x log x

x(loglog x)k-1
(k - 1) ! (log x)
loglog x≤k≤log x
k+2
≪
k+2
(k-1) logloglog x- loglog x-k log k+k+O(log k)
x
log x
loglog x≤k≤log x
(k-1) logloglog x- loglog x-k log k+2 k+2+O(log k)
≪ x
x

log x
loglog x≤k≤log x
log x
-k log k
≪x
x

log x
≪x
loglog x≤k≤log x
(loglog x) (logloglog x)
x
.
log x
(29)
These complete the estimate. ■
Lemma 11.
Let x ∈  be a large number, let q > x1/2 log-1 x be an integer, and let ω(q) > loglog x. Then,
ρ(q) logm q

q
1-
(loglog x) (logloglog x)
≪ x
2 log x
, m ≥ 1.
(30)
Proof: Let ρk (q) =   n ≤ x1/2 : n2 + 1 ≡ 0 mod q and ω(q) = k , and observe that

x1/2-ϵ <q≤x,
ω(q)>loglog x
ρ(q) ≤

1<q≤x,
k>loglog x
ρk (q) =

k>loglog x
ρk (x) πk (x) = W (t).
(31)
12 | Quadratic Primes
Now, the finite sum is estimated as follows:
ρ(q) logm q

1/2-ϵ
x
<q≤x,
ω(q)>loglog x
logm q
≤
q

1<q≤x,
ω(q)>loglog x
q
x logm
=
1
t
t
d W (t)
x
logm t
=
t
x
W (t) - 
1
logm t
t
′
(32)
W (t) d t
1
logm x
≪
x
W (t) .
Apply Lemma 10 to complete the estimate. ■
7. Small Distances Between Powers Of Integers
The Catalan conjecture claims that the sequence of integers powers 1, 22 , 23 , 32 , 24 , 52 , 33 , 25 , 62 , 72 , 26 , 34 , 102 , ... has
only one pair of consecutive powers, namely, 23 and 32 . This result, usually expressed by the Diophantine equation
x p - yq = 1, p, q ≥ 2, was proved a few years ago, detailed information is widely available in the literature. The proofs of
several special cases were established long ago.
Lemma 12. (Lebesgue-Nagell) For any exponent n ∈ ℕ, the Diophantine equation x2 + 1 = yn has no nonzero integers
x, y ∈ ℤ solutions.
For even n = 2 m, m ≥ 1 the proof is trivial. And for odd n = 2 m + 1, the equation has genus g > 1, so it has a finite
number of solutions. An algebraic proof appears in various places in the literature, [SC08, p. 9]. The generalized LebesgueNagell equation has been completely solved for a small range of parameters.
Lemma 13. ([BS06]) The Diophantine equation x2 + d = yn , with n ≥ 3, and 1 ≤ d ≤ 100, has at most eight integer
solutions x, y ∈ ℤ.
A complete table of solutions appears in [BS06]. A few of these equations have no solutions at all, for example, x 2 + 1 = yn
and x2 + 3 = yn . On the other direction, x2 + 28 = yn has the most solutions:
(x, y, n) = (6, 4, 3), (22, 8, 3), (225, 37, 3), (2, 2, 5), (6, 2, 6), (10, 2, 6), (22, 2, 9), (362, 2, 17) .
Quadratic Primes | 13
Both Lemmas 12 and 13 immediately imply that the quadratic arithmetic progressions  n 2 + d : n ∈ ℤ  have bounded
numbers of prime powers, that is, n2 + d = pv : n ∈ ℤ, v ≥ 2. Specifically, the finite sums
Λn2 + 1

n2 +1= pv ≤ x, v≥2
n
=0
Λn2 + d
and
= O(1)

log n
n2 +d= pv ≤ x, v≥2
n
(33)
log n
for 1 ≤ d ≤ 100. Employing standard analytical method, it can be shown that these finite sums are bounded by a constant.
But the algebraic proofs of the Lebesgue-Nagell equation give exact answers.
The number of solutions of the more general case n2 + d = pv , n ∈ ℤ, v ≥ 2, and d ≠ 0 constant, will be of interest in
proof for primes of the form n2 + d = p : n ∈ ℤ.
8. Asymptotic Formula
Let x ∈  be a large number, and let π f (x) =  { f (n) ≤ x : f (n) = p is prime } be the corresponding counting function of
the prime numbers defined by the irreducible polynomial f (x) ∈ ℤ[x]. For the specific case f (x) = x2 + 1, the expected
asymptotic formula has the form
π f (x) =  1 p≥3
1
-1
p-1
p
x
log x
+o
x
log x
= (1.3727 ...)
x
log x
+o
x
log x
,
(34)
see [RG04, p. 7], [NW00, p. 342]. A lower bound for the counting function is given below.
Lemma 14.
Let x ≥ 1 be a large number, then   p = n2 + 1 ≤ x : p is prime  ≫ x1/2  log x.
Proof: By means of Theorem 1, the lower bound can be obtained by partial summation:
  p = n2 + 1 ≤ x : p is prime  ≫ 
2
n +1 ≤ x
where R(t) = ∑n2 +1 ≤ t Λn2 + 1 n
-1
log n  . ■
Λn2 + 1 n
n
log n
log n
log n
x1/2
=
2
t
log t
d R(t) ≫
x1/2
log x
,
(35)
14 | Quadratic Primes
9. Some Related Results
The problem investigated here can also be viewed as a special case m = d, 1 ≤ d ≤ 100, of the results in [FI97 and [FI98]
for primes of the forms p = n2 + m2 , and p = n2 + m4 , m, n ∈ ℤ.
9.1 Counting Functions For Quadratic Primes In Two Variables
Theorem 15. ([FI98]) Let f (r, s) = r2 + s4 ∈ ℤ[r, s], (an absolutely irreducible polynomial over the integers), and let
x ≥ 1 be a large number. Then,
 Λn2 + m4  =
2
4
n +m ≤ x
1
where the constant κ = ∫0
1 - t4 d t = Γ(1 / 4)2  6
4κ
x3/4 1 + O
π
loglog x
log x
,
(36)
2 π .
Theorem 15. ([FI97]) Let x ≥ 1 be a large number, and let χ ≠ 1 be a character mod 4. Then,
 Λ(n) Λn2 + m2  = 2  1 2
2
p≥2
n +m ≤ x
χ( p)
( p - 1) ( p - χ( p))
x+O
x
log x
.
(37)
9.2 The PSI Function For Quadratic Polynomials
The well known psi function asymptotic formula
ψ(n) = log lcm(1, 2, 3, ..., n) = n + O(n / log n)
(38)
was extended to the quadratic arithmetic progression. The new result claims the following.
Theorem 16. ([CJ10[, [RZ13]) For θ < 4 / 9, and f (x) = x2 + 1, the expression
ψ f (n) = log lcm12 + 1, 22 + 1, 32 + 1, ..., n2 + 1 = n log n + B n + On  logθ n,
(39)
where the constant
B = γ-1-

log 2
2
-
p≠2
-1
 log
p
p-1
p
= -.0662756342 ... .
(40)
Quadratic Primes | 15
9.3 Largest Prime Factors And Density Of Primitive Factors
Let P(n) denotes the largest prime factor of the integer n ≥ 1. Some of the history of the largest prime factor Pn 2 + 1 , and
in general P( f (n)) for any irreducible polynomial f (x) ∈ ℤ[x], is discussed in [NW00, p. 345].
Theorem 17. ([ES90]) Let f (x) ∈ ℤ[x] be irreducible of degree d = deg( f ) > 1. Then there exists a constant c 1 > 0
such that for x > x1 ( f ),
P  f (n) > xe
ec1 (loglogx)
1/3
...
.
(41)
n≤x
Theorem 18.
satisfies
([DI82]) For infinitely many integers n ∈ ℤ, the largest prime factor of the polynomial f (x) = x 2 + 1
Pn2 + 1 ≥ n1.202468... .
(42)
A factor m  An of an integer sequence An , n ≥ 1, is called a primitive factor if
(1) m  An for n ≤ n0 , and (2) m  An for some n > n0 .
Define the density function ρ( f ) = limx→∞  { n ≤ x : f (n) has a primitive factor } / x.
Theorem 19. ([EH08]) Let x ≥ 1, and f (x) = x2 + d be an irreducible polynomial. Then, it satisfies
x ≪ ρn2 + d ≪ x.
(43)
10. Some Problems
Problem 1. Prove that there are infinitely many twin quadratic primes n2 + 1, n2 + 3 as n⟶∞. The sequence has the
initial pairs (101, 103), (197, 199), (5477, 5479), (8837, 8839), ... . This problem is discussed in [NW00, p. 342].
Problem 2. Determine the minimal and maximal gaps of two consecutive quadratic primes
pk = n2 + 1, pk+1 = m2 + 1, m, n ≥ 1. The minimal gap is (n + 2)2 + 1 - n2 + 1 = 4 n + 2 ≥
pk . And the average gap is
also large:
x  π f (x) = x  c0 x1/2  log x + ox1/2  log x = c0 x1/2 log x + ox1/2 log x,
where c0 = 1.3727 ... is a constant. Thus, this problem is quite different from the linear primes
{ 2, 3, 5, 7, 11, 13, 17, 19, ... }, which has an average of x / π(x) = log x + o(log x).
For n2 + 1 ≤ 10 000, the sequence of primes is
2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, ...,
(44)
16 | Quadratic Primes
and the sequence of gaps is
3, 12, 20, 96, 60, 144, 176, 100, 620, 304, 1316, 220, 1220, 1120, 1580, 1044, 736 , ... .
Problem 3. The exact values of several series associated with primes producing polynomials are known to be transcendental numbers:
1

n≥0
n2 + 1
=
π e2 π + 1
1
+ ,
2 e2 π + 1
2
1

n≥0
n2 + 3
e2 π
3
3 e2 π
3
π
=
2
+1
1
+ ,
6
+1
etc.,
(45)
see [SR74, p. 189-199]. Does the transcendence of the series implies that the sequence of denominators contain infinitely
many primes?
Problem 4. Prove that there are infinitely many primes 2 p2 + 1 as the prime p⟶∞. The sequence
a p2 + 1, a ≥ pϵ , and ϵ > 0, has infinitely many primes as the prime p ⟶∞, this sequence is constructed in [MK09].
Problem 5. Determine the least primitive root of the sequence of quadratic primes n 2 + 1 as n⟶∞. The primitive roots of
some quadratic sequences are studied in [AS13].
Problem 6. Show that the function f (s) = ∑n≥1 Λn2 + 1 n-s has a pole at s = 1. Hint: Let
f (s) = lim 
x→∞
n≤x
Λn2 + 1
ns
,
(46)
then use an approximation as in the proof of Theorem 1.
Problem 7. Prove that the class number h(d) of the quadratic numbers field  d  satisfies h(d) > 1 for infinitely many
discriminants d = n2 + 1 ≥ 5. This problem was considered in [MN88].
Quadratic Primes | 17
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